1 problem found
Let \(\F(x)\) be the cumulative distribution function of a random variable \(X\), which satisfies \(\F(a)=0\) and \(\F(b)=1\), where \(a>0\). Let \[ \G(y) = \frac{\F(y)}{2-\F(y)}\;. \] Show that \(\G(a)=0\,\), \(\G(b)=1\,\) and that \(\G'(y)\ge0\,\). Show also that \[ \frac12 \le \frac2{(2-\F(y))^2} \le 2\;. \] The random variable \(Y\) has cumulative distribution function \(\G(y)\,\). Show that \[ { \tfrac12} \,\E(X) \le \E(Y) \le 2 \E(X) \;, \] and that \[ \var(Y) \le 2\var(X) +\tfrac 74 \big(\E(X)\big)^2\;. \]
Solution: \begin{align*} && G(a) &= \frac{F(a)}{2-F(a)}\\ &&&= 0 \tag{\(F(a)= 0\)}\\ \\ && G(b) &= \frac{F(b)}{2-F(b)} \\ &&&= \frac{1}{2-1} = 1 \tag{\(F(b)=1\)}\\ \\ && G'(y) &= \frac{F'(y)(2-F(y))+F(y)F'(y)}{(2-F(y))^2} \\ &&&= \frac{2F'(y)}{(2-F(y))^2} \geq 0 \tag{\(F'(y) \geq 0\)} \end{align*} \begin{align*} && 0 \leq F(y)\leq1\\ \Leftrightarrow&& 1\leq 2-F(y) \leq 2\\ \Leftrightarrow &&1 \leq (2-F(y))^2 \leq 4\\ \Leftrightarrow && 1 \geq \frac{1}{(2-F(y))^2} \geq \frac14 \\ \Leftrightarrow && 2 \geq \frac{2}{(2-F(y))^2} \geq\frac12 \end{align*} \begin{align*} && \mathbb{E}(Y) &= \int_a^b y G'(y) \d y \\ &&&= \int_a^b y F'(y) \underbrace{\frac{2}{(2-F(y))^2}}_{\in [\frac12, 2]} \d y \\ &&&\leq 2 \E[X] \\ &&&\geq \frac12 \E[X]\\ \\ && \E[Y^2] &\leq 2\E[X^2] \\ && \E[Y^2] &\geq \frac12\E[X^2] \\ \\ \Rightarrow && \var[Y] &= \E[Y^2]-\E[Y]^2 \\ &&& \leq 2 \E[X^2] - (\tfrac12\E[X])^2 \\ &&&= 2 \var[X] + \tfrac74(\E[X])^2 \end{align*}